This chapter eases us from informal ideas into formal mathematics. We motivate that move, develop more formal approaches to structures we’ve already seen, acclimatize ourselves to the formalism, and see what we get out of it. We argue that formal mathematical language and notation are to do with both efficiency and abstraction, as concise notation can help us to package up multiple concepts into a single unit that we can then handle as a new object. One example is repeated addition turning into multiplication, and then repeated multiplication turning into exponentiation. We revisit metrics and express them more formally. We also cover some basic formal logic, which is how formal mathematics is built up securely, including logical implication, converses, and logically equivalent ways of stating them. We also revisit modular arithmetic and show how to express it more formally.